Hydraulic conductivity in spatially varying media—a pore-scale ......Geophys. J. Int. (2011) 184, 1167–1179 doi: 10.1111/j.1365-246X.2010.04893.x GJI Marine geosciences and applied

Geophys. J. Int. (2011) 184, 1167–1179 doi: 10.1111/j.1365-246X.2010.04893.x

GJI

Mar

ine

geos

cien

ces

and

appl

ied

geop

hysics

Hydraulic conductivity in spatially varying media—a pore-scaleinvestigation

J. Jang,1 G. A. Narsilio2 and J. C. Santamarina31School of Civil and Environmental Engineering, 790 Atlantic Drive, Georgia Institute of Technology, Atlanta, GA 30332, USA. E-mail: [email protected] of Infrastructure Engineering, The University of Melbourne, Parkville, VIC 3010, Australia3School of Civil and Environmental Engineering, 790 Atlantic Drive, Georgia Institute of Technology, Atlanta, GA 30332, USA

Accepted 2010 November 14. Received 2010 August 28; in original form 2009 November 13

S U M M A R YThe hydraulic conductivity can control geotechnical design, resource recovery and waste dis-posal. We investigate the effect of pore-scale spatial variability on flow patterns and hydraulicconductivity using network models realized with various tube size distributions, coordinationnumber, coefficient of variation, correlation and anisotropy. In addition, we analyse flow pat-terns to understand observed trends in hydraulic conductivity. In most cases, the hydraulicconductivity decreases as the variance in pore size increases because flow becomes gradu-ally localized along fewer flow paths; as few as 10 per cent of pores may be responsible for50 per cent of the total flow in media with high pore-size variability. Spatial correlation reducesthe probability of small tubes being next to large ones and leads to higher hydraulic conduc-tivity while focused fluid flow takes place along interconnected regions of high conductivity.A pronounced decrease in tortuosity is observed when pore size and spatial correlation in theflow direction are higher than in the transverse direction. These results highlight the relevanceof grain size and formation history dependent pore size distribution and spatial variability onhydraulic conductivity, related geo-process and engineering applications.

Key words: Numerical solutions; Hydrology; Microstructure; Permeability and porosity.

1 I N T RO D U C T I O N

The hydraulic conductivity k depends on the size of pores, theirspatial distribution and connectivity. These pore-scale characteris-tics are defined by grain size distribution and formation history. Inturn, hydraulic conductivity controls fluid invasion, flow rate andpore fluid pressure distribution. Consequently, hydraulic conduc-tivity affects storativity, effective stress and mechanical stability,plays a critical role in geotechnical design, determines contam-inant migration and the selection of remediation strategies, de-fines the limits for resource recovery (oil production and resid-ual oil saturation, gas extraction from hydrate bearing sediments,methane recovery from coal bed methane, non-isothermal fluidflow in geothermal applications), and is a central parameter in thedesign of waste disposal strategies, from nuclear waste to CO2sequestration.

In this study, we investigate the effect of pore-scale spatial vari-ability on macroscale hydraulic conductivity using network models,following the pioneering work by Fatt (1956a,b,c). The main advan-tage of network models resides in their ability to capture pore-scalecharacteristics within a physically sound upscaling algorithm torender macroscale properties relevant to the porous medium. Net-works can be generated either by assuming an idealized regulargeometry, by adopting physically representative networks that cap-

ture the porous structure (Bryant et al. 1993) or by mapping thepore structures measured by high resolution tomographic technicsonto a network structure (Dong & Blunt 2009, see also Al-Raoush& Wilson 2005; Narsilio et al. 2009). Network model results areconsistent with experimentally obtained values of permeability (Al-Kharusi & Blunt 2007; Al-Kharusi & Blunt 2008). The approachhas been used to upscale a wide range of pore-scale phenomenasuch as viscous drag, capillarity, phase change (e.g. ice or hydrate)and mineral dissolution. Consequently, network models have beenused to study multiphase flow (Valvatne 2004; Al-Kharusi & Blunt2008), wettability effects in multiphase flow (Suicmez et al. 2008),fine migration and clogging (Kampel et al. 2008), mineral disso-lution (Hoefner & Fogler 1988; Fredd & Fogler 1998), pressure-induced pore closure (David 1993), CO2 sequestration (Kang et al.2005), liquid or gas diffusion through porous media (Laudone et al.2008; Mu et al. 2008), drying and unsaturation (Prat 2002; Surasaniet al. 2008), the effect of flow localization on diffusion (Bruderer& Bernabé 2001) and resource recovery such as methane produc-tion from hydrate bearing sediments (Tsimpanogiannis & Lichtner2003; Tsimpanogiannis & Lichtner 2006). Furthermore, pore-scalenetwork models have been coupled to continuum models to con-duct field-scale simulations of complex processes such as clogging,reactive flow and non-Darcian flow near well-bores (Balhoff et al.2007).

C© 2011 The Authors 1167Geophysical Journal International C© 2011 RAS

Geophysical Journal International

1168 J. Jang, G. A. Narsilio and J. C. Santamarina

The first part of the manuscript summarizes previous studies.Then, we provide a detailed description of the numerical model, re-port statistical results in terms of equivalent hydraulic conductivityand compare trends against known and analytically derived lowerand upper bounds.

2 VA R I A B I L I T Y I N H Y D R AU L I CC O N D U C T I V I T Y — P R E V I O U S S T U D I E S

Hydraulic conductivity can vary by more than 10 orders of magni-tude, from very low values in montmorillonitic shale to high val-ues in gravels and boulders. Hydraulic conductivity varies widelyeven for a given material. The coefficient of variation, defined asthe ratio between the standard deviation and the mean, can rangefrom 100 to 800 per cent for both natural sediments (Libardi et al.1980; Warrick & Nielsen 1980; Cassel 1983; Albrecht et al. 1985;Duffera et al. 2007) and remolded sediments (Benson 1993; Benson

& Daniel 1994). Data are typically log-normal distributed so thatx = log(k/[k]) is Gaussian, where [k] captures the dimensions of k(Freeze 1975; Hoeksema & Kitanidis 1985).

The correlation length L is the distance where the spatial autocor-relation decays by 1/e ∼= 0.368. The correlation length for hydraulicconductivity ranges from less than a meter to hundreds of meters. Itis typically longer in the horizontal plane than in the vertical direc-tion, in agreement with layering and weathering patterns (Ditmarset al. 1988; Bjerg et al. 1992; DeGroot 1996; Lacasse & Nadim1996).

The equivalent hydraulic conductivity keq of spatially varyingmedia reflects the distribution of individual values ki, their spatialcorrelation and flow conditions. Available close-form solutions aresummarized in Table 1. In particular, the equivalent hydraulic con-ductivity keq is (1) the harmonic mean of individual ki values in 1-Dsystems, (2) the geometric mean in 2-D media and (3) higher thanthe geometric mean when seepage in 3-D systems can take place

Table 1. Equivalent hydraulic conductivity-mixture models and bounds.

Equivalent, keq Assumptions/comments References

ka =N∑

i=1ti ki

/N∑

i=1ti

Arithmetic mean—parallelParallel stratified mediaUpper bound

Wiener (1912)

kh =N∑

i=1ti

/N∑

i=1

tiki

Harmonic mean—SeriesPerpendicular stratified media(1-D flow)Lower bound

kg =(

N∏1

ki

)1/N Geometric meanLognormal ki distribution/isotropicmedia (2-D flow)

Warren & Price (1961)

keq = kαa k1−αh(

α = D − 1D

) Statistically homogeneous andisotropic. Weighted average ofWiener bounds

Landau & Lifshitz(1960)

ka − f1 f0(k1 − k0)2

k0(D − f0) + k1 f0 ≤ keq ≤ ka −f1 f0(k1 − k0)2

k1(D − f1)2 + k0 f1Based on a model constructed ofcomposite spheres. Isotropic binarymedium. Uniform flow

Hashin & Shtrikman(1962)

if f0 ≥ 0.5 ⇒ keq ≥ kacif f0 ≤ 0.5 ⇒ keq ≤ kacif f0 = 0.5 ⇒ keq =

√k1k0,

where kac = 12

[( f1 − f0)(k1 − k0) +

√( f1 − f0)2(k1 − k0)2 + 4k1k0

]

Isotropic 2-D random two phasemosaic medium. Uniform flow.

Matheron (1967)

if f0 ≥ 0.5 ⇒ keq ≤ km

km =f1k0k1 + f0ka

√k0(2ka − k0)

f1m∗ + f0√

k0(2ka − k0)if f0 ≤ 0.5 ⇒ keq ≥ km

km = k0k1 f0ka + f1√

k0(2m∗ − k0)f1k0k1 + f1m∗

√k0(2m∗ − k0)

Note: keq is the equivalent hydraulic conductivity; f 0 and f 1 are the fractions of the medium with hydraulic conductivity k0 and k1, where k1 > k0, D is thespace dimension (i.e. 1, 2 or 3); m∗ = f1k0 + f0k1.

C© 2011 The Authors, GJI, 184, 1167–1179Geophysical Journal International C© 2011 RAS

Hydraulic conductivity – spatial variability 1169

through multiple alternative flow paths. The equivalent hydraulicconductivity in these three cases can be computed in terms of thegeometric mean kg and the variance σ 2 in log(k/[k]), as captured inthe following expressions (Gutjahr et al. 1978; Dagan 1979—referto Table 1):

keq = kh = kg[1 − (σ 2log k/2)] (1-D system) (1)

keq = kg (2-D system) (2)

keq = kg[1 + (σ 2log k/6)] (3-D system). (3)

More complex systems have been studied using equivalent con-tinuum numerical methods. Those results show that (1) flow ratedecreases as the coefficient of variation COV(k) increases and (2)the mean hydraulic conductivity in correlated fields is higher thanin uncorrelated fields with the same coefficient of variation COV(k)(Griffiths & Fenton 1993; Griffiths et al. 1994; Griffiths & Fenton1997).

3 N E T W O R K M O D E L S

Network models consist of tubes connected at nodes and can beused to simulate fluid flow through pervious materials. Volume canbe added at nodes to reproduce various conditions (Reeves & Celia1996; Blunt 2001; Acharya et al. 2004). The flow rate through atube q [m3 s−1] is a function of fluid viscosity η [Ns m−2], tuberadius R [m], tube length �L [m] and pressure difference betweenend nodes �P [N m−2]

q = π R4

8 η �L�P = α �P tube equation − Poiseuille, (4)

where α = πR4/(8η�L) under isothermal condition, constant vis-cosity and constant tube radius. Mass conservation requires that thetotal flow rate into a node equals the total flow rate out of the node∑

qi = 0 node equation. (5)Eqs (4) and (5) can be combined to determine the pressure at acentral node Pc as a function of the pressure at neighbouring nodesPi.

Pc =∑

αi Pi∑αi

. (6)

If all α-values are equal, eq. (6) predicts Pc = (Pa + Pb + Pr + Pl)/4.It is worth noting that this equation is identical to the first-ordercentral finite difference formulation of Laplace’s field equation.

Eq. (6) is written at all internal nodes to obtain a system of linearequations which can be captured in matrix form

AP = B, (7)where the matrix A is computed with tube conductivities α, P isthe vector of unknown pressures at internal nodes and the vector Bcaptures known boundary pressures. The vector P can be recoveredas P = A−1B. Once fluid pressures Pi are known at all nodes,the global flow rate Q through the network is obtained by addingthe flow rate q (eq. 4) in all tubes that cross a plane normal to theflow direction. The equivalent network hydraulic conductivity in thedirection of the prescribed external pressure gradient is calculatedfrom the computed flow rate Q and the imposed pressure gradientbetween inlet and outlet boundaries. Insightful information is gainedby analysing prevailing flow patterns within the networks as will beshown later in this manuscript.

Networks are realized with pre-specified statistical characteris-tics. We control the coefficient of variation in tube size, spatialcorrelation and isotropy to generate networks with different tubesize distribution (monosized, bimodal or log-normal distributed)spatially uncorrelated or correlated and isotropic or anisotropic.Every realization is identified according to these three qualifiers.

Pore size R is log-normally distributed in sediments. Mercuryintrusion porosimetry data for a wide range of soils and effectivestress conditions show that the standard deviation in σ [ln(R/[μm])]is about 0.4 ± 0.2. Examples of statistical distributions used inthis study are shown in Fig. 1. Throughout the manuscript, the log-normal distribution of pore cross sectional area is used in termsof R2, that is, log(R2/[R]2) where [R] indicates unit of R. Tube R2

values are generated as R2 = 10a where a is a set of Gaussiandistributed random numbers with given standard deviation. ValuesR2 are scaled to satisfy the selected mean value. While we assume

Figure 1. Schematic of typical distribution of R2 in spatially varying fields.(a) Bimodal distribution of tubes (used in Figs 2 and 3) when fraction ofsmall tubes is 20 per cent. The relative size of large to small tube radii is(RL/RS)4 = 103. (b) Two distributions of tube size R2 with the same μ(R2)but different standard deviation used in Figs 4, 6 and 7. As the coefficientof variation increases, the distribution of R2 is skewed to the right. (c)Distributions of R2 used in Fig. 5. Note that σ (R2) of two sets of tube sizedistribution with different μ(R2) are adjusted to have same COV(R2).



log-normal distribution for network generation, we analyse resultsand global trends in terms of the mean and standard deviation ofR2, that is μ(R2) and σ (R2) for each realization.

The computer code is written in MATLAB. The run time for eachrealization in a 2.4 GHz processor is ∼40 min. The reported studywas conducted using a stack of dual-core computers.

4 S T U D I E D C A S E S — N U M E R I C A LR E S U LT S

Network models are used herein to extend previous studies on theeffect of spatial variability and anisotropy on hydraulic conductivity.

Numerical results are presented next. Simulation details are listedin the corresponding figure captions.

4.1 Bimodal distribution—effect of coordinationnumber and bounds

Consider a bimodal distribution made of large and small tubes ofrelative size RL/RS = 5.62 so that their conductivity ratio is kL/kS =103 for constant tube length (refer to eq. 4). 20 spatially randomlyarranged networks are generated for each fixed fraction of smalltubes. 2-D networks with coordination number, cn = 4, 6 and 8and 3-D networks with coordination number cn = 6 are used to

Figure 2. Effect of coordination number cn on equivalent hydraulic conductivity in bimodal distribution kmix normalized by the hydraulic conductivity in thefield composed of only large tubes kL. Each point is the average value of 20 realizations. Bimodal distribution of tubes. The relative size of large L to small Stube radii is (RL/RS)4 = 103. 2-D network model: 50 × 50 nodes, 4900 tubes and cn = 4 (circle)/cn = 6 (triangle)/cn = 8 (square). 3-D network model: 15 ×15 × 15 nodes, 9450 tubes and cn = 6 (diamond).

Figure 3. Computed equivalent hydraulic conductivity in bimodal distribution kmix normalized by the hydraulic conductivity in the field composed of onlylarge tubes kL, models and bounds as a function of the fraction of small tubes. Points represent the maximum (square), average (triangle) and minimum (circle)values of 20 realizations at each fraction of small tubes. Bounds and models are described in Table 1. 2-D network model: 50 × 50 nodes, 4900 tubes, bimodaldistribution of tubes, relative size of large L to small S tube radii (RL/RS)4 = 103 and coordination number cn = 4.



investigate the effect of coordination number on flow conditions.Computed hydraulic conductivities are averaged for the 20 realiza-tions and plotted in Fig. 2 where the mean value is normalized bythe hydraulic conductivity of the network model made of large tubesonly, kmix/kL.

Results in Fig. 2 show that network conductivity range in threeorders of magnitude from kmix/kL = 0.001 to 1.0 in agreement with

the size ratio (RL/RS)4 = 103. Hydraulic conductivity values in-crease as the coordination number increases. There is a pronounceddecrease in flow rate where large tubes cease to form a percolat-ing path. Percolation thresholds (readily identified in linear–linearplots—see also Hoshen & Kopelman 1976) decrease as coordinationnumbers increase; results are consistent with reported percolationthresholds for various networks: 2-D-honeycomb (fraction of small

Figure 4. Equivalent hydraulic conductivity in uncorrelated tube network kdist normalized by the hydraulic conductivity for the monosized tube network kmonoas a function of the coefficient of variation of R2. Each point is a single realization. All realizations have the same μ(R2). 2-D network model: 50 × 50 nodes,4900 tubes and cn = 4 (empty triangle), cn = 6 (empty square), cn = 8 (empty circle). 3-D network model: 15 × 15 × 15 nodes, 9450 tubes and cn = 6 (soliddiamond). Shaded areas show arithmetic ka, geometric kg, harmonic kh mean of 2-D and 3-D system and analytical solution keq of 3-D system (eq. 3).

Figure 5. Anisotropic conductivity: equivalent hydraulic conductivity in uncorrelated and distributed tube network kdist normalized by the hydraulic conductivityfor the monosized tube network kmono as a function of coefficient of variation of R2. Normalized hydraulic conductivities are obtained at different values ofthe ratio between the mean tube size parallel and transverse to the flow direction (RP/RT)2. Each point is the average value of 20 realizations (using same setof tube sizes, but different spatial distribution). For clarity, results for intermediate sequences are shown as shaded area. Normalized hydraulic conductivitiesin series of parallel and parallel of series circuits are also obtained. 2-D network model: 50 × 50 nodes, 4900 tubes, cn = 4 and log-normal distribution of R2.



tubes = 0.65), 2-D-square (0.5), 2-D-triangular (0.35) and 3-D-simple cubic arrangement (0.25) (Stauffer & Aharony 1992; Sahimi1994).

Analytical solutions for equivalent hydraulic conductivity andlower and upper bounds summarized in Table 1 are compared tonumerical results in Fig. 3. The normalized mean hydraulic con-ductivity for 20 realizations using 2-D networks with cn = 4 fol-lows the Matheron’s mixture model. All simulation results are be-tween Wiener’s and Hashin and Shtrikman’s upper and lower bounds(Table 1). Hashin and Shtrikman bounds incorporate the dimension-ality of the system resulting in 3-D bounds that are shifted towardshigh keq values compared to 2-D bounds. Overall, numerical andanalytical results point to higher value of hydraulic conductivitywith a larger number of alternative flow paths.

4.2 Coefficient of variation in random networks

We explore next the effect of variance in R2 by creating 2-D and 3-Dnetworks with the same nominal mean μ(R2). Network statistics,mean μ(R2), standard deviation σ (R2) and coefficient of variationCOV(R2) are evaluated for each realization. Note that R2 distribu-tions are skewed towards higher values as the coefficient of variationincreases (Fig. 1b) even though they all have the same μ(R2).

The conductivity of a given realization kdist is normalized bythe conductivity kmono of the network made of all equal size tubes,that is, R2 = μ(R2) and COV(R2) = 0. The normalized hydraulicconductivity kdist/kmono decreases as the coefficient of variation ofR2 increases (Fig. 4) (see similar results in Bernabé & Bruderer1998). The normalized arithmetic, geometric and harmonic meanscomputed for each network are shown as shaded areas on Fig. 4.The range in normalized hydraulic conductivities for 2-D cn =4 networks coincides with the shaded band of geometric meanscomputed for all networks. Computed hydraulic conductivity valuesfor 3-D cn = 6 and 2-D cn = 6 networks are the same as the rangeobtained using eq. (3) and confirm the applicability of the close-form solutions.

These trends result from the increased probability of large tubesbecoming surrounded by smaller tubes, that is, there is an increasedprobability of finding a small tube along every potential flow pathwith increasing coefficient of variation COV(R2). This effect is morepronounced when the coordination number decreases because thereare fewer alternative flow paths; in other words, network models withhigh coordination number are less sensitive to variation in pore sizeCOV(R2) because a higher number of alternative flow paths developin high connectivity condition. Flow patterns are analysed in detaillater in this manuscript.

Figure 6. Correlated field. (a) Equivalent hydraulic conductivity in isotropic uncorrelated and correlated tube network kcor normalized by the hydraulicconductivity for the monosized tube network kmono as a function of the coefficient of variation of R2. (b) Coefficient of variation of the equivalent hydraulicconductivities as a function of the coefficient of variation of R2. The correlation length L is reported relative to the specimen size. Each point stands for theaverage of 100 realizations. 2-D network model: 40 × 40 nodes, 3120 tubes, cn = 4 and log-normal distribution of R2.



4.3 Anisotropic, uncorrelated networks

When tubes parallel to the predominant fluid flow direction aremonosized RP (‘parallel tubes’), the flow rate is proportional toR4P and the distribution of tube size transverse to flow direction RT(‘transverse tubes’) does not affect the global flow rate because thereis no local gradient or fluid flow transverse to the main flow direc-tion. This is not the case when tubes parallel to the flow directionare of different size, that is, not monosized.

Let’s consider log-normal distributions for the size R2P and R2T of

both parallel and transverse tubes. We select different mean valuesμ(R2P) = μ(R2T) and adjust standard deviations σ (R2P) and σ (R2T) sothat both parallel and transverse tubes have the same coefficient ofvariation COV(R2).

Results in Fig. 5 show that the normalized hydraulic conductivitydecreases as the coefficient of variation COV(R2) increases whenμ(R2P)/μ(R

2T) > 1. However, the hydraulic conductivity may ac-

tually increase when transverse tubes are of high conductivity asshown by the μ(R2P)/μ(R

2T) = 10−2 case: fluid flows along trans-

verse tubes until it finds parallel tubes of high conductivity, mostlywith R2P > μ(R

2P).

Two extreme networks of ‘series-of-parallel’ and ‘parallel-of-series’ tubes provide upper and lower bounds to the numerical results(shown as lines in Fig. 5). When the ratio of (RP/RT)2 is larger than10−1, the network responds as a parallel combination of tubes inseries. When the ratio of (RP/RT)2 is smaller than 10−1, pressure ishomogenized along the relatively large transverse tubes, as capturedin the series-of-parallel bound.

4.4 Spatial correlation in pore size—isotropic networks

Spatial correlation in pore size upscales to the macroscale hydraulicconductivity in unexpected ways. The methodology followed in thisstudy starts with a set of tubes with fixed μ(R2) and COV(R2). Then,we use the same set of tubes to generate 100 randomly redistributedspatially uncorrelated networks and other three sets of 100 isotropi-cally correlated networks with correlation lengths L/D = 5/39, 15/39

Figure 7. Effect of anisotropic correlation on equivalent hydraulic conductivity in an anisotropically correlated tube network kCOV>0 normalized by thehydraulic conductivity for the monosized tube network kmono. Three sets of tubes different COV(R2) are generated and used to form correlated fields ofdifferent anisotropic correlation length. LP and LT are the correlation lengths parallel and transverse to flow direction. D is the length of medium perpendicularto the flow direction. In the range between LP/LT = 0.01–1, LP = 2D/39 fixed and LT changes from 2D/39 to 30D/39. In the range between LP/LT = 1–100,LT = 2D/39 fixed and LP changes from 2D/39 to 30D/39. Each point is an average of 20 realizations. 2-D network model: 40 × 40 nodes, 3120 tubes, cn = 4and log-normal distribution of R2.



and 30/39 (where L is correlation length and D is the network sizetransverse to the overall flow direction) and for different COV(R2).We use the method by Taskinen et al. (2008) to create correlatedfields.

Hydraulic conductivities are numerically computed for all net-works kcor. For comparison, the hydraulic conductivity kmono is eval-uated for a network of equal size tubes, that is, COV(R2) = 0. Thenormalized mean hydraulic conductivity kcor/kmono computed usingthe 100 realizations for each COV(R2) is plotted versus COV(R2) inFig. 6a. The normalized mean conductivity decreases with COV(R2)in all cases in agreement with Fig. 4, but it is higher in correlatedthan in uncorrelated networks. Note that the variance from the meantrend also increases with COV(R2) and it is exacerbated by spatialcorrelation L/D (Fig. 6b).

4.5 Spatial correlation in pore size—anisotropic networks

To gain further insight into the previous results, we study the effectof anisotropy in correlation length following a similar approach,but in this case we distinguish the correlation length parallel tothe overall flow direction LP from the correlation length transverseto the overall flow direction LT. The isotropic case is created withLP/D = LT/D = 2/39 so that LP/LT = 1.0. High correlation parallelto the flow direction is simulated by increasing LP/D, while highcorrelation transverse to the flow direction is imposed by increasingLT/D. The study is repeated for three sets of R2 with COV(R2) =0.5, 1.4 and 2.8.

Average hydraulic conductivity values (based on 20 realizations)are normalized by the hydraulic conductivity kmono of the networkmade of equal size tubes. Results in Fig. 7 show that the normalizedhydraulic conductivity kCOV>0/kmono increases as spatial correlationparallel to the flow direction LP/LT increases and it may even exceedthe conductivity of the monosized tube network in highly anisotropicnetworks with very high LP/LT values. Otherwise, variation in tubesize COV(R2) has a similar effect reported previously: an increasein COV(R2) causes a decrease in hydraulic conductivity (see Figs 4and 6). Overall, results in Fig. 7 point to pore-scale flow conditionssimilar to those identified in Fig. 5.

5 D I S C U S S I O N

Numerical results show the evolution of percolation in bimodalsystem (Figs 2 and 3) and the decrease in hydraulic conductivitywith increasing variance in pore size while the mean value of poresize remains constant. This is observed for all types of networktopology (Fig. 4) and in both spatially correlated and uncorrelatednetworks (Fig. 6). The only exception to this trend is found inhighly anisotropic porous media in the direction that favours fluidflow (Figs 5 and 7).

To facilitate the visualization of flow patterns, we compute tubeflow rates (eq. 4) and represent tubes with lines of thickness pro-portional to flow rate (additional plotting details are noted in figurecaptions). Fig. 8(a) shows flow patterns in bimodal distribution net-works made of different fractions of small tubes. Flow localizes

Figure 8. Analysis of flow pattern in network model of bimodal distribution of R2 (2-D cn = 4 – Percolation occurs when the fraction of small tubes is 0.5.Refer to Figs 2 and 3 for simulation details). (a) Flow intensity in each tube of the network of different fraction of small tubes. The change of flow pattern ineach fraction of small tubes is well detected. The arrow indicates the predominant fluid flow direction. (b) Fraction of tubes tube50 per cent responsible for 50per cent of total conductivity. The fraction total-tube50 per cent is the summation of small-tube50 per cent and large-tube50 per cent.



along dominant flow channels when the fraction of small tubes is50 per cent, which is near the percolation threshold for this network(2-D cn = 4). Few flow paths are responsible for the global conduc-tivity in networks where the fraction of either small or large tubes is∼50 per cent (Fig. 8a�); conversely, multiple flow paths contributeto the global conductivity in networks made of a majority of eithersmall or large tubes (Fig. 8a� and �).

The fraction of parallel tubes, which conducts 50 per cent ofthe total flow, tubes50 per cent, quantifies this observation (Fig. 8b).The values is tubes50 per cent = 50 per cent when all tubes are of thesame size, either large or small, which means flow is homogeneous.Fluid preferentially flows along the large tubes so that large tubesare responsible for 50 per cent of the total flow until the fraction ofsmall tubes exceeds ∼65 per cent. The participation of small tubesstarts to increase above the large-tube percolation threshold (0.5 –Point � in Fig. 8b). In general, flow always seeks the larger tubes.

Distributed tube diameters exhibit a similar response. Most par-allel tubes contribute to total flow when the coefficient of varia-

tion of R2 is low (Fig. 9a�). Flow becomes gradually localized asCOV(R2) increases and fewer channels contribute to global flow(tubes50 per cent in Fig. 9c). Consequently, hydraulic conductivity de-creases as shown earlier (Figs 4 and 6). The main effect of spatialcorrelation is to channel flow along interconnected regions of highconductivity (compare Figs 9a and b, see also Bruderer-Weng et al.2004 for the effect of different correlation lengths on flow chan-nelling).

Flow patterns in anisotropic networks are shown in Figs 10and 11 for 18 realizations with different degrees of anisotropyμ(R2P)/μ(R

2T), spatial correlation LP/LT and tube size variability

COV(R2). The number of parallel tubes responsible for 50 percent of the total flow is included in Fig. 12 for all cases. Signif-icant flow takes place along transverse tubes when transverse tubesare much more conductive than parallel tubes μ(R2P) μ(R2T)(Fig. 10a), or when there is high transverse correlation LP/LT

1 (Fig. 11a—upper bound was labelled ‘series-of-parallel’ config-uration in Fig. 5). On the other hand, there is virtually no flow

Figure 9. Analysis of flow pattern in network model of log-normal distribution of R2 (refer to Figs 4 and 6 for simulation details). (a) Flow intensity in eachtube in spatially uncorrelated network. (b) Flow intensity in each tube in spatially correlated networks. Thickness of line represents the intensity of flow rate.(c) Fraction of tubes tube50 per cent responsible for 50 per cent of total conductivity.



Figure 10. Analysis of flow patterns in anisotropic networks made of tubes with the same mean size μ(R2) but different variance in size as captured in COV(R2)(refer to Fig. 5 for simulation details). Anisotropy ratios: (a) μ(R2P)/μ(R

2T) = 0.01, (b) μ(R2P)/μ(R2T) = 1.0, (c) μ(R2P)/μ(R2T) = 100. The arrow indicates the

global flow direction. The line thickness used to represent the tubes is proportional to the flow intensity in each tube. Tortuosity values τ are shown for eachcase.

along transverse paths when parallel tubes are much larger thanthe transverse tubes μ(R2P) � μ(R2T) (Fig. 10c) or when there ishigh longitudinal correlation LP/LT � 1 (Fig. 11c); in these cases,flow localizes along linear flow paths and global conductivity islimited by the smallest tubes along their longitudinal paths (thirdrow in Figs 10 and 11—referred to the ‘parallel-of-series’ boundin Fig. 5). Therefore, the number of parallel tubes responsible formost of the flow decreases with increasing COV(R2) in this case aswell.

Let’s define the network tortuosity factor as the ratio τ =(NCP/Nhom)2 between the total number of tubes in the backbone ofthe critical path NCP and the number of tubes in a straight streamlineparallel to the global flow direction Nhom (details in David 1993). Acritical path analysis in terms of tube flow rate is used to computethe tortuosity factors for fluid flow (see Bernabé & Bruderer 1998).Figs 10 and 11 show flow patterns and associated tortuosity values.In agreement with visual patterns, there is a pronounced decrease in

tortuosity when μ(R2P) � μ(R2T) in anisotropic uncorrelated fields(Fig. 10) or when LP/LT � 1 in anisotropic correlated fields withhigh COV(R2) (Fig. 11).

Spatial correlation reduces the probability of small tubes beingnext to large ones and leads to more focused channelling of fluidflow through the porous network. This can be observed by visualinspection of cases shown in Fig. 11 in comparison to the corre-sponding ones in Fig. 10.

Simple geometrical analyses show that the distance between ad-jacent pore centres is 2R for simple cubic packing and face-centredcubic packing and

√1.5R for tetrahedral packing, where R is the

grain radius (see also Lindquist et al. 2000). However, the constanttube length assumption made in this study is only an idealization forreal sediments (Bryant et al. 1993). For example, the distance be-tween adjacent pore centres in Fontainebleau and Berea sandstonesranges from 20 to 600 μm with most tube lengths between 130 and200 μm (Lindquist et al. 2000; Dong & Blunt 2009).



Figure 11. Analysis of flow pattern in anisotropically correlated networks made of three sets of tube areas with different COV(R2) (refer to Fig. 7 for simulationdetails). Anisotropy ratios: (a) LP/LT = 1/15, (b) LP/LT = 1/1 and (c) LP/LT = 15/1. The arrow indicates the global flow direction. The line thickness used torepresent the tubes is proportional to the flow intensity in each tube. Tortuosity values τ are shown for each case.

Figure 12. Fraction of tubes tubes50 per cent carrying 50 per cent of the total flux as a function of (a) coefficient of variation of R2 in anisotropically uncorrelatedfield and (b) the ratio of parallel to transverse correlation length LP/LT.



While pore-to-pore distance varies in real sediments, we note thatthe hydraulic conductivity of tubes is much more dependent on theradius than on the tube length (see eq. 4). Therefore, the imposedvariability in tube radius causes variability in tube conductivity qthat could equally capture tube length variability. Clearly, variationsin tube length would imply a non-regular network topology.

6 C O N C LU S I O N S

Grain size and formation history dependent pore size distributionand spatial variability determine the hydraulic conductivity, im-miscible fluid invasion and mixed fluid flow, resource recovery,storativity and the performance of remediation strategies. Numeri-cal simulations with porous networks permit the study of pore-sizedistribution, spatial correlation and anisotropy on hydraulic con-ductivity and flow patterns in pervious media.

In most cases, the hydraulic conductivity decreases as the vari-ance in pore size increases because flow becomes gradually local-ized along fewer flow paths. As few as 10 per cent of pores maybe responsible for 50 per cent of the total flow in media with highpore-size variability. The equivalent conductivity remains withinHashin and Shtrickman bounds.

Spatial correlation reduces the probability of small pores beingnext to large ones. There is more focused channelling of fluid flowalong interconnected regions of high conductivity and the hydraulicconductivity is higher than in an uncorrelated medium with thesame pore size distribution.

The equivalent hydraulic conductivity in anisotropic correlatedmedia increases as the correlation length parallel to the flow direc-tion increases relative to the transverse correlation. The hydraulicconductivity in anisotropic uncorrelated pore networks is boundedby the two extreme ‘parallel-of-series’ and ‘series-of-parallel’ tubeconfigurations. Flow analysis shows a pronounced decrease in tor-tuosity when pore size and spatial correlation in the flow directionare higher than in the transverse direction.

While Poiseuille flow defines the governing role of pore sizeon hydraulic conductivity, the numerical results presented in thismanuscript show the combined effects of pore size distribution andvariance, spatial correlation and anisotropy (either in mean pore sizeor in correlation length). In particular, results show that the properanalysis of hydraulic conductivity requires adequate interpretationof preferential flow paths or localization along interconnected highconductivity paths, often prompted by variance and spatial correla-tion. The development of flow localization will impact a wide rangeof flow related conditions including the performance of seal layersand storativity, invasion and mixed fluid flow, contaminant migra-tion and remediation, efficiency in resource recovery, the formationof dissolution pipes in reactive transport and the evolution of finemigration and clogging.

A C K N OW L E D G M E N T S

Support for this research was provided by the National ScienceFoundation and by DOE/NETL Methane Hydrate Project undercontract DE-FC26–06NT42963.

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